Chapter 2: Q8P (page 74)

Evaluate each of the following in $x + iy$ form, and compare with a computer solution $i^{2 / 3}$ .

Short Answer

Expert verified

$The x + iy form of the given equation, i^{2 / 3} . w_{1} = \frac{1}{2} + i \frac{\sqrt{3}}{2} w_{2} = \frac{1}{2} - i \frac{\sqrt{3}}{2} w_{3} = - 1$

Step by step solution

Given Information.

The given expression is $i^{2 / 3}$ .

Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of x+iy in which x is the real part and y is the imaginary part.

Convert in polar form.

$C o n s i d e r, w = i^{2 / 3} W r i t e t h e g i v e n n u m b e r i n p o l a r f o r m . w = e^{[I n {(i)}^{2 / 3}]} U s e t h e \log a r i t h m p r o p e r t y . w = e^{[\frac{2}{3} I n (i)]} w = e^{(\frac{2}{3} (I n r + (θ + 2 n π) i))} w = e^{(\frac{2}{3} (\frac{π}{2} + 2 n π) i)} w = e^{((\frac{π}{3} + \frac{4 n π}{3}) i)} W h e r e n = 0, \pm 1, \pm 2, \pm 3, . . . . .$

Write in the form of x+iy.

Use the definition of so In and $θ = \frac{π}{2}$ because the number is pure imaginary.

It’s the third root so it has three values.

Put the values.

Put n=0

$w_{1} = e^{(π / 3)} w_{1} = \frac{1}{2} + i \frac{\sqrt{3}}{2}$

Put n=1

$w_{2} = e^{(5 π / 3)} w_{2} = \frac{1}{2} - i \frac{\sqrt{3}}{2}$

Put n=-1

$w_{3} = e^{(- π)} w_{3} = - 1$

Therefore, x+iy form of the given equation $i^{2 / 3}$ .

$w_{1} = \frac{1}{2} + i \frac{\sqrt{3}}{2} w_{2} = \frac{1}{2} - i \frac{\sqrt{3}}{2} w_{3} = - 1$

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Evaluate each of the following in $x + iy$ form, and compare with a computer solution $i^{2 / 3}$ .

Short Answer

Step by step solution

Given Information.

Meaning of rectangular form.

Convert in polar form.

Write in the form of x+iy.

One App. One Place for Learning.

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Evaluate each of the following in x+iyform, and compare with a computer solution i2/3.

Short Answer

Step by step solution

Given Information.

Meaning of rectangular form.

Convert in polar form.

Write in the form of x+iy.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Famous Physicists

Work Energy and Power

Dynamics

Solid State Physics

Kinematics Physics

Study anywhere. Anytime. Across all devices.

Evaluate each of the following in $x + iy$ form, and compare with a computer solution $i^{2 / 3}$ .